My Math Forum limit of arctan(x/m)

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April 27th, 2018, 02:03 AM   #1
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limit of arctan(x/m)

please check the given limit is correct or not. as the value of arctanx or tan inverse x cannot exceed π/2, but its limit written here is π for m<O.
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 April 27th, 2018, 02:08 AM #2 Senior Member   Joined: Oct 2009 Posts: 405 Thanks: 140 You are right, it doesn't make any sense.
 April 27th, 2018, 03:33 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,313 Thanks: 2447 Math Focus: Mainly analysis and algebra It depends on the definition of $\arctan x$. If we define $$-\frac\pi2 < \arctan x < \frac\pi2$$ then you are correct, but if we define $$0 < \arctan x < \pi$$ then the book is correct. Thanks from topsquark
 April 27th, 2018, 06:22 AM #4 Global Moderator   Joined: Dec 2006 Posts: 19,042 Thanks: 1618 Does the book allow $s$ to be negative? Thanks from topsquark
April 27th, 2018, 10:52 PM   #5
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Quote:
 Originally Posted by v8archie It depends on the definition of $\arctan x$. If we define $$-\frac\pi2 < \arctan x < \frac\pi2$$ then you are correct, but if we define $$0 < \arctan x < \pi$$ then the book is correct.
but, if we define arctanx to (-π,π) ,
will arctanx remain a function?
it will have three values for one x.
Help me if I am wrong.
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April 27th, 2018, 10:56 PM   #6
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Quote:
 Originally Posted by Shariq Faraz please check the given limit is correct or not. as the value of arctanx or tan inverse x cannot exceed π/2, but its limit written here is π for m
Quote:
 Originally Posted by skipjack Does the book allow $s$ to be negative?
s can be any complex or real number, but greater than 0.

April 27th, 2018, 11:01 PM   #7
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Quote:
 Originally Posted by Shariq Faraz but, if we define arctanx to (-π,π) , will arctanx remain a function? it will have three values for one x. Help me if I am wrong.
Don't define it on $(-\pi,\pi)$ then. I suggest you consider the interval that I mentioned instead.

 April 27th, 2018, 11:18 PM #8 Global Moderator   Joined: Dec 2006 Posts: 19,042 Thanks: 1618 If $s$ isn't real, "greater than zero" doesn't make sense. If $s$ is real and positive, the original question is effectively about a one-sided limit. It's possible for arctan(0) to be defined as $\pi$ (or some other multiple of $\pi$), but it's usually defined as 0. The situation with the arccot function is rather different... it's much easier to find differences of opinion about arccot(0). Thanks from topsquark

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